(x0,y0) is a source is there exists a neighbourhoodU of (x0,y0) s.t. for any p∈U, φt(p) leaves U as t increases.

(x0,y0) is a sink if there exists a neighbourhood S of (x0,y0) s.t for any p∈S, φt(p) approaches (x0,y0) at an exponential rate as t increases, In this case (x0,y0) is an example of an attracting set and its basin of attraction is given by: B = ∪t≤0φt(S).

where the jacobian at the origin: (-α 0, 0 -β)(P,Q), reflects the hyperbolic nature of the equilibrium. The linearization about the origin is given by:

(Ṗ,Q̇) = (-α 0, 0 -β)(P,Q).

It is easy to see for the linearised system that:

Es = {(P,Q) | Q = 0},

Eu = {(P,Q) | P = 0}

(1) is the invariant stable subspace and (2) is the invariant unstable subspace.

To state how the saddle point structure is inherited by the nonlinear system, we state the results of the stable and unstable manifold theorem for hyperbolic equilibria:

First, consider two intervals of the coordinate axis containing the following as follows: IP = { -ε < P < ε } and IQ = { -ε < Q < ε } for some small ε>0. A neighbourhood of the origin is constructed by taking the cartesian product of these two intervals:

Bε = { (P,Q) ∈ R2 | (P,Q) ∈ IP x IQ }.

The stable and unstable manifold theorem for hyperbolic equilibrium points states the following:

There exists two Cr curves given by the graph as a function of the P and Q variables (resp.):

1. Q = S(P), P∈IP,

2. P = S(Q), Q∈IQ.

The first curve has three important properties: it passes through the origin, i.e. S(0) = 0; It is tangent to ES at the origin; it is locally invariant. The curve satisfying these three properties is unique and is referred to as the local stable manifold of the origin, denoted by:

Wsloc((0,0)) = {(P,Q) ∈ Bε | Q = S(P) }.

The second curve has three important properties: it passes through the origin, i.e. U(0) = 0; it is tangent to EU at the origin; it is locally invariant.. For these reasons it is referred to as the local unstable manifold of the origin, and it is denoted by: